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Math Help - series 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7...

  1. #1
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    series 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7...

    Hello!

    First, thanks for all the wonderful help during this course of Real Analysis! This website is amazing!

    I have to determine whether the series: 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7 + 1/8 + 1/9 - 1/10 - 1/11 ... converges or diverges

    I assume I have to do a rearrangement to show this, but I don't know how.
    Please, give me some hints

    thanks
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by inthequestofproofs View Post
    Hello!

    First, thanks for all the wonderful help during this course of Real Analysis! This website is amazing!

    I have to determine whether the series: 1 - 1/2 -1/3 + 1/4 + 1/5 - 1/6 -1/7 + 1/8 + 1/9 - 1/10 - 1/11 ... converges or diverges

    I assume I have to do a rearrangement to show this, but I don't know how.
    Please, give me some hints

    thanks
    Alternating series test - Wikipedia, the free encyclopedia

    Edit: Sorry, I read your question wrong.
    Last edited by chiph588@; April 29th 2010 at 12:53 PM.
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  3. #3
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    Okay so that is the sum to infinity of (-1)^{n+1}\left(\frac{1}{n}\right) so look to see if this converges absolutely
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  4. #4
    MHF Contributor chisigma's Avatar
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    The series can be written as...

    S= 1 + \sum_{n=1}^{\infty} (-1)^{n} (\frac{1}{2n} + \frac{1}{2n+1}) (1)

    .. so that is an alternating sign series whose terms are decreasing with n and tend to 0 if n tends to infinity: the series converges...

    Kind regards

    \chi \sigma
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  5. #5
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    BTW: you can only rearrange the terms if the series converges absolutely.
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